7.4 Partial Fractions Finding a partial fraction decomposition is the opposite operation of finding a common denominator. We are tearing a rational expression apart into its component pieces. Decomposition of N(x)/D(x) into Partial Fractions: 1. Divide if improper: If N(x)/D(x) is an improper fraction, divide the denominator into the numerator to obtain a polynomial plus a proper fraction. 2. Factor the denominator: Completely factor the denominator into factors of the form m px q and 2 n ax bx c where the quadratic is irreducible. 3. Linear factors: For each factor of the linear factors, the partial fraction decomposition must include the following sum of m fractions 1 2 2 ... m m A A A px q px q px q 4. Quadratic Factors: For each quadratic factor, the partial fraction decomposition must include the following sum of n fractions 1 1 2 2 2 2 2 2 ... n n n Bx C Bx C Bx C ax bx c ax bx c ax bx c Examples: Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. 1. 2 2 4 3 x x x 2. 2 3 2 3 2 4 11 x x x x